# Lojban Wave Lessons/19

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 Lojban Wave Lessons: Foreword | ← Lesson 18 | Lesson 19 | Lesson 20 →

## Lesson 19: Numbers

When learning a language, one of the things which are usually taught very early on is how to count. This really makes little sense, because it's not necessary to know numbers if you don't know how to speak about those things to which they apply. This is partly the reason why I have left it for lesson number nineteen. The other reason is that while the numbers themselves are easy to learn, how they apply to sumti can get very confusing indeed. That, however, we will save for a later lesson.

Before learning the words themselves, you should know that numbers do not have any internal grammar. This means that any row of number words (henceforth referred to as a "number string") are treated identically to any other number string to the Lojban grammar, even if the string makes no sense. Therefore, one can never answer unambiguously whether a number construct makes sense or not. There are, however, intended ways of using the number words, and confusion will probably result if you deviate from the standard.

Learning all the number words of Lojban are way beyond the scope of this lesson, so you will only be introduced to what is normally used in text. The wide range of Lojban mathematical cmavo are called mekso (Lojban for "mathematical expression"), and is widely disregarded because of its complexity and questionable advantage over so-called bridi math.

Let's begin with the ordinary Lojban numbers, from zero to nine:

Notice how the vowels are alternating (with the exception of no), and how no consonant is used for two digits. In order to express numbers higher than nine, the numbers are just strung together:
• vo mu ci – four hundred and fifty three (453)
• pa no no no no ten thousand (10 000)
There is also a question-digit, which is used like other fill-in-the-blank question words. It's xo. The answer to such a question may be just the relevant digit(s) by itself, or they can be numerical constructs, as shown later.
• ci xo xo xo – "Three thousand and how many?" (3???)
xo = question digit – use like any other digit to ask for the correct digit.
The experimental word xo'e is sometimes used to mean an unspecified, elliptical digit. Its definition is not official, though.
• ci xo'e xo'e xo'e – Three thousand and something
xo'e = elliptical digit.
Since all number strings are treated grammatically the same, one might also answer several digits to one xo'e Furthermore, there is also a set of hexadecimal digits A through F. By default, Lojban operates in base 10, but when using hexadecimal digits, it can be safely assumed that you use base sixteen:
 dau fei gai jau rei xei vai 10(A) 11(B) 12(C) 13(D) 14(E) 14(E) 15(F)

Yes, I know there are two words for E. The official one is rei (all three-letter cmavo beginning with x is experimental). xei was invented to avoid confusion with re.

The base can be explicitly stated using ju'u: Any number before ju'u the number being spoken of, any number after is the base of the number:

• dau so fei no ju'u pa re – A9B0 in base 12 (notice here that base 12 is always in decimal. It is possible to permanently change the base you speak in, but since it has never been used in practice, it has not been standardized how one should do it)

Fractions are also useful to learn how to express. They are usually expressed via a decimal point, pi.

pi = Decimal point (or point in whichever base you are talking in)

pa re pi re mu – twelve point two five (12.25).

Like in mathematics, when no number string is placed before or after pi, zero is assumed.

Related to this, the number separator pi'e is used to separate numbers, either to separate digits when speaking in a base larger than sixteen, or when a decimal point is not applicable, for instance, when talking about time in hours, minutes, seconds:

pa so pi'e re mu pi'e no ju'u re ze – Nineteen, twenty-five, zero in base 27 (JP0 base 27)

re re pi'e vo bi – twenty-two, fourty eight (22:48)

There is also a range of number words which are not mathematically exact but rather subjective or relative. The behaviors of these words are almost exactly like the behavior of digits, except they cannot be combined to make bigger numbers the way digits can:

 ro so'a so'e so'i so'o so'u all almost all most many some few

When combined with any of the digits, these words are assumed to give a second verdict about the size of the number:

mu bi so'i sai – Fifty eight, which is really many.

They should therefore not be placed in the middle of a number string. When placed after pi, they are assumed to convey the size of a fraction:

• pi so'u – a small part of it
• pi so'o – some of it
• pi so'i – a large part of it
• pi so'e – most of it
• pi so'a – almost all of it

These are some hightly subjective numbers - they work just like the previous ones.

 du'e mo'a rau too many too few enough

The following five are context-based numbers – these work like the previous ones, with the exception that they take the next number in order to assign them meaning:

 da'a su'e su'o za'u me'i all except n At most n At least n more than n less than n

If no number string follow them, one is assumed.

so'i pa re da'a mu – Many, which is twelve, which is all but five.

The two last number words you should know have slightly more complicated grammar:

ji'i = number rounding or number approximation

When ji'i is placed before a number, the entire number is approximated:

ji'i ze no za'u rau ju'o – "About seventy, which is more than enough, certainly

Placed in the middle of the number, only the following digits are non-exact. At the end of a number, it signifies that the number has been rounded off.

ki'o = Number comma - separates digits within one string; Thousands.

It is not incidential that ki'o sounds like kilo. At its simplest, ki'o is used to separate three digits at a time in large numbers, much like commas are used in English:

pa ki'o so so so ki'o bi xa ze – 1,999,867

If less than three digits are put before a ki'o, the digits are assumed to be the least significant ones, and zeros are assumed to fill in the rest:

vo ki'o ci bi ki'o pa ki'o ki'o – 4,038,001,000,000

ki'o is used similarly after a decimal point.

That concludes the common Lojban numbers themselves. How they apply to sumti is a science in itself, and we leave that for lesson twenty-two. Now we focus on how these numbers can be used in a bridi.

A string of number words by themselves are grammatical, since they can act as an answer to a xo-type of question. In this case, however, they cannot be considered part of any bridi. In general, if numbers fill part of a bridi, they do so in one of two forms: Pure numbers and quantifiers. We will return to quantifiers in a later lesson. For now, we will look at pure numbers.

A pure number is any row of number words prefixed with li. This makes a sumti directly from the number, and refers to the mathematical concept of, for instance, the number six. Its famyma'o is lo'o

li = convert number/mekso-expression to sumti.
lo'o = famyma'o: end convert number/mekso-expression to sumti.

These pure sumti are usually what fills the x2 of brivla such as mitre or cacra

mitre = x1 is x2 metres in dimension x3 by standard x4
cacra = x1 is x2 hours in duration (default 1) by standard x3

Try to translate the following:

le ta nu cinjikca cu cacra li ci ji'i u'i nai

Answer: (sigh) That flirting has been going on for around three hours.

How do you count to three in Lojban?

Answer: li pa li re li ci

The last thing we'll go through in this lesson is the words of the selma'o MAI and those of MOI.

MAI only contains two words, mai and mo'o. Both of these convert any number string to an ordinal, which has the grammar of attitudinals. Ordinals are used to divide a text into numbered segments, like chapters or parts. The only difference between mai and mo'o is that mo'o quantifies larger subdivisions of text, allowing you to divide a text on two different levels, for example enumerating chapters with mo'o and sections with mai. Notice that these as well as the MOI take any number string directly, without any need for li.

mai = Lower-order ordinal marker: Convert number to ordinal.
mo'o = Higher order ordinal marker: Convert number to ordinal.

There are five MOI, and they all convert any number string to selbri. We'll take them one at a time:

moi = Convert number n to selbri: x1 is the n'th member of set x2 by order x3

Example:

 la lutcimin ci moi lo'i ninmu pendo be mi le su'u lo clani zmadu cu lidne lo clani mlecaLui-Chi Min is third among my female friends by the order: The more tall ones precedes the less tall ones.

When specifying a sequence, it is widely understood that if a ka-abstraction (lesson thirty) is used as a sumti, the members of the set are ordered from the one with most of the property to the one with less of the property, so the x3 of the following sentence could have been shortened to lo ka clani.

lidne = x1 is before x2 in sequence x3
clani = x1 is long in dimension x2 by standard x3
zmadu = x1 exceeds x2 in property/aspect x3 by amount x4
mleca = x1 is less than/is less characterized than x2 by property/aspect x3 by amount x4
mei = Convert number n to selbri: x1 is the mass formed from the set x2, which has the n members of x3

Notice here that x3 are supposed to be individuals, x2 a set and x1 a mass.

What would mi ci mei mean?

Anwer: We are group of three.

si'e = Convert number n to selbri: x1 is n times x2

Example: le vi plise cu me'i pi pa si'e lei mi cidja be ze'a lo djediThis apple here is less than one tenth of my food for one day

Please note that the definition of si'e when looked up will tell you that it's "x1 is an nth of x2", instead of "x1 is n times x2". But people only use it as I have defined it, so the definition in the dictionaries will probably change.

cu'o = Convert number n to selbri: x1 has n probability of occurring under conditions x2

Example:

 lo nu mi mrobi'o cu pa cu'o lo nu mi denpa riAn event of me dying has probability 1 under the conditions: I wait for it = Me dying is completely certain if I wait long enough.
denpa = x1 waits for x2, being in state x3 until resuming/doing x4
va'e = Convert number n to selbri: x1 is at the n'th position on the scale x2

Example:

 li pa no cu ro va'e la torinon10 is the highest value on the Torino-scale.
 Lojban Wave Lessons: Foreword | ← Lesson 18 | Lesson 19 | Lesson 20 →